Problem: The warming or cooling rate of a drink is proportional to the difference between the ambient temperature $T_a$ and the current temperature $T$ of the drink. Which equation describes this relationship? Choose 1 answer: Choose 1 answer: (Choice A) A $T(t)=k(T_a-T)$ (Choice B) B $T(t)=\dfrac{k}{(T_a-T)}$ (Choice C) C $\dfrac{dT}{dt}=k(T_a-T)$ (Choice D) D $\dfrac{dT}{dt}=\dfrac{k}{(T_a-T)}$
Solution: The current temperature of the drink is denoted by $T$. The rate of change of the temperature is represented by $T'(t)$, or $\dfrac{dT}{dt}$. Saying that the rate of change is proportional to something means it's equal to some constant $k$ multiplied by that thing. That thing, in our case, is the difference between the ambient temperature, $T_a$ and the current temperature, $T$, of the beverage. In conclusion, the equation that describes this relationship is $\dfrac{dT}{dt}=k(T_a-T)$.